Course for international guest/part time students

Faculty

Faculty of Science

Organization

TTK Department of Geometry

Code

riegeo1u0um17gm

Title

Riemannian geometry 1 (p)

Usual semester

Autumn

ECTS

2

Language

Learning outcomes

Knowledge: Knowledge of basic concepts, results and methods in the field. Ability: Application of knowledge in the field, understanding of interrelationships and problem solving. Attitude: Desire to improve mathematical knowledge and to learn as much as possible, and to apply knowledge as widely as possible. Autonomy and responsibility: Formulate and analyze mathematical questions independently and evaluate the limits of their applicability responsibly.

Course content

Vector bundles Linear connection on a vector bundle Parallel transport along a curve The curvature tensor Riemannian manifold, Levi–Civita covariant derivative Exponential map on a Riemannian manifold Variational formulas for arc length Jacobi fields along a geodesic Conjugate points The index form associated with a geodesic The completeness problem for Riemannian manifolds, the Hopf–Rinow theorem Rauch comparison theorems Manifolds with nonpositive Gaussian curvature, the Cartan–Hadamard theorem

Assessment method

(written or oral) exam plus term mark

Bibliography

lecture notes

Recommended bibliography

D. Gromoll, W. Klingenberg, W. Meyer: Riemannsche Geometrie im Grossen, Lecture Notes in Mathematics, 55, Springer-Verlag, 1975. J. Cheeger, D. Ebin: Comparison theorems in Riemannian geometry, North-Holland, 1975.

Programmes of the course